发表机构
KAIST; Institute for Basic Science (IBS)（韩国科学技术院; 基础科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对大次数一般曲线，在Brill--Noether轨迹维数预期且特定上同调消失条件下，给出所有分次Betti数的闭式公式，并确定完整Betti表及Boij--Söderberg系数的渐近行为。
AI 中文摘要
设$C$为亏格$g$、gonality为$k$的光滑射影复曲线，$L$为$C$上的极丰沛线丛。当$L$的次数足够大时，Koszul上同调群$K_{p,q}(C,L)$的消失与非消失性此前已被确定，但分次Betti数$\kappa_{p,q}(C, L)$的精确值在很大程度上仍属未知。本文中，当Brill--Noether轨迹$W_k^1(C)$具有预期维数且$H^1(C, L \otimes \omega_C^{-1})=0$时，我们给出了所有分次Betti数$\kappa_{p,q}(C, L)$的显式闭式公式。因此，当$\deg L \geq 4g-3$或当$\deg L \geq 3g-3$且$L$为一般时，我们确定了一般曲线的完整Betti表。我们还显式计算了控制渐近纯性的截面环$R(C, L)$的Boij--Söderberg系数，并证明了其余系数的最终单调性：对于超椭圆曲线它们递减，而在相关Brill--Noether轨迹的自然一般约化假设下它们递增。
英文摘要
Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $κ_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $κ_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes ω_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $°L \geq 4g-3$ or when $°L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--Söderberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.
Comments25 pages
